How Java s FloatingPoint Hurts Everyone Everywhere


 Kerrie Cunningham
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1 How Java s FloatingPoint Hurts Everyone Everywhere by Prof. W. Kahan and Joseph D. Darcy Elect. Eng. & Computer Science Univ. of Berkeley Originally presented 1 March 1998 at the invitation of the ACM 1998 Workshop on Java for High Performance Network Computing held at Stanford University This document: or July 30, :28 pm Work in Progress Subject to Supersession Page 1
2 Pages Topics 3 Abstract 49 Overview: Java has evolved to target markets to which its initial design decisions are illsuited. 10 Pure Java s Two Cruel Delusions, promises Java cannot keep Example: Complex Arithmetic Classes; should misplotted fluid flows be exactly reproducible? 16 Example: Faster Matrix Multiply too valuable to forego for unneeded exact reproducibility SelfDiscipline, Reproducibility, Controllability 19 Java purports to fix what ain t broken in Floatingpoint Exceptions; Algebraical Completion; lack of Flags makes Java s FloatingPoint Dangerous Misconceptions about Floatingpoint Example: Disassociate Catastrophic from Cancellation ; Computation as a Web 31 An old Rule of Thumb is wrong because of misconceptions about Precision and Accuracy Why so many still believe this wrong rule of thumb; another counterexample What s wrong with it (and another counterexample); how it got into so many programming languages What to do instead; Four Rules of Thumb for best use of modern floatingpoint hardware 44 Example: Angle at the eye; old KernighanRitchie C semantics are safer than Java s Three Williams contend for Java s numerics, it should copy old KernighanRitchie C semantics Example: 3dimensional rectilinear geometry; Crossproducts work better as matrix products Overloaded operators; Neat solutions for nearestpoint problems, turned into numerical junk by Java s floatingpoint, work well in KernighanRitchie C Dynamic Directed Rounding Modes; Debugging Numerical Instability Example: Needlelike triangles area and angles IEEE 754 Double Extended reduces the risk of chagrin, conserves monotonicity, but not in Java. Three floatingpoint formats run fast; the widest is valuable for Example: Cantilever calculation; Iterative refinement s accuracy improves spectacularly more than 11 bits The cheaper machines would always get better results but for Java s and Microsoft s intransigence How to support extraprecise arithmetic; anonymous indigenous ; Optimizations by the Compiler 81 Conclusions: Java s floatingpoint hurts Java vs. J++, so repair Java s floatingpoint soon. July 30, :28 pm Work in Progress Subject to Supersession Page 2
3 Abstract: Java s floatingpoint arithmetic is blighted by five gratuitous mistakes: 1. Linguistically legislated exact reproducibility is at best mere wishful thinking. 2. Of two traditional policies for mixed precision evaluation, Java chose the worse. 3. Infinities and NaNs unleashed without the protection of floatingpoint traps and flags mandated by IEEE Standards 754/854 belie Java s claim to robustness. 4. Every programmer s prospects for success are diminished by Java s refusal to grant access to capabilities built into over 95% of today's floatingpoint hardware. 5. Java has rejected even mildly disciplined infix operator overloading, without which extensions to arithmetic with everyday mathematical types like complex numbers, intervals, matrices, geometrical objects and arbitrarily high precision become extremely inconvenient. To leave these mistakes uncorrected would be a tragic sixth mistake The following pages expand upon material presented on Sunday morning 1 March 1998 partly to rebut Dr. James Gosling s keynote address Extensions to Java for Numerical Computation the previous morning (Sat. 28 Feb.); see his For a better idea of what is in store for us in the future unless we can change it, see and July 30, :28 pm Work in Progress Subject to Supersession Page 3
4 We agree with James Gosling about some things like Some kind of infix operator overloading will have to be added to Java. Some kind of Complex class will have to be added to Java. Some changes to the JVM are unavoidable. 95% of the folks out there are completely clueless about floatingpoint. ( J.G., 28 Feb ) ( Maybe more than 95%?) and disagree with him about other things like A proposal to enhance Java s numerics would split the Java community into three parts: 1. Numerical Analysts, who would unanimously be enthusiastically FOR it, 2. Others, who would be vehemently AGAINST it, and 3. Others who wouldn t care. ( J.G., 28 Feb ) Actually, Numerical Analysts would be as confused as everyone else and even more divided. Complex arithmetic like Fortran s? That s not the best way. The C9X proposal is better. Loose Numerics? Sloppy numerics! IEEE 754 DoubleExtended supported properly is better. and many more July 30, :28 pm Work in Progress Subject to Supersession Page 4
5 To cure Java s numerical deficiencies, we too propose to modify it but not the way Gosling would modify it. We call our modified Java language Borneo. Borneo s design was constrained to be Upward Compatible with Java : Compiling Java programs with Borneo semantics should leave integer arithmetic unchanged and should change floatingpoint arithmetic at most very slightly. Any old Java class already compiled to bytecode should be unable to tell whether other bytecode was compiled under Java s semantics or Borneo s. Borneo is designed to require the least possible change to the Java Virtual Machine ( JVM ) that can remedy Java s floatingpoint deficiencies. Borneo adds to Java as little infix operator overloading, exception flag and trap handling, control over rounding directions and choice of precisions as is essential for good floatingpoint programming. If you wish not to know about them, don t mention them in your program. For more information about Borneo : For more information about FloatingPoint : What follows is NOT about Borneo. What follows explains why Java has to be changed. By Sun. Urgently. July 30, :28 pm Work in Progress Subject to Supersession Page 5
6 + + Anne and Pete use the same program. But they do not use the same platform. See Pat. How? How can this be? Pat wrote one program. It can run on all platforms. They have 100% Pure Java. It works with the platforms Pat used 100% Pure Java (TM) they have. to write the program. Anne and Pete are happy. Run program, run! They can work. Work, work, work! mul ti plat form lan guage no non Java (TM) code write once, run a ny where (TM) 100% Pure JAVA Pure and Simple This parody of puffery promoting 100% Pure Java for everyone everywhere filled page C6 in the San Franisco Chronicle Business Section of Tues. May 6, It was paid for and copyrighted by Sun Microsystems. Behind Sun s corporate facade must have twinkled a wicked sense of humor. July 30, :28 pm Work in Progress Subject to Supersession Page 6
7 Whom does Sun expect to use Java? Everybody. Everybody falls into one of two groups: 1. A roundup of the usual suspects These numerical experts, engineers, scientists, statisticians, are used to programming in C, Fortran, Ada, or to using programs written in those languages. Among their programs are many that predate IEEE Standard 754 (1985) for Binary FloatingPoint Arithmetic; these programs, many written to be Portable to the computers of the 1970s, demand no more from floatingpoint than Java provides, so their translation into Java is almost mechanical. 2. Everybody else 95% of the folks out there are completely clueless about floatingpoint. ( J.G., 28 Feb ) Their numerical inexpertise will not deter clever folks from writing Java programs that depend upon floatingpoint arithmetic to perform parts of their computations: Materials lists and blueprints for roofing, carpentry, plumbing, wiring, painting. Numerically controlled machine tools and roboticized manufacturing, farming and recycling. Customizable designs for homebuilt furniture, sailboats, light aircraft, gokarts, irrigation. Navigation for sailboats, light aircraft and spaceships while their pilots doze at the wheel. Economic and financial forecasts, estimated yield on investments, and portfolio management. Predictions of supply and demand, predictive inventory management, justintime delivery. There is no end to this list. July 30, :28 pm Work in Progress Subject to Supersession Page 7
8 Q & A about selling computing to Everyone Everywhere: What would happen to the market for automobiles if transmissions and chokes were not automatic, and if brakes and steering were not not powerassisted? Would all drivers be dextrous and strong, or would there be fewer cars and more chauffeurs as in the good old days? What if standards for vehicular bodystrength, lights, brakes, tires, seatbelts, airbags, safetyglass, were relaxed? Would cheaper cars and trucks compensate us for the cost of caring for more cripples? Are such questions irrelevant to our industry? What will happen to the market for our computer hard and software if we who design them fail to make them as easy to use as we can and also robust in the face of misuse? Misuse is unavoidable. Our industry s vigor depends upon a vast army of programmers to cope with innumerable messy details some of which, like floatingpoint, are also complicated; and In every army large enough, someone fails to get the message, or gets it wrong, or forgets it. Most programmers never take a competent course in Numerical Analysis, or else forget it. Over 95% of the folks out there are completely clueless about floatingpoint. ( J.G., 28 Feb ) Amidst an overabundance of Java Beans and Class Libraries, we programmers usually hasten to do our job without finding the information we need to cope well with floatingpoint s complexities. Like Coleridge s Ancient Mariner afloat in Water, water every where, nor any drop to drink we are awash in (mis and dis)information. To filter what we need from the worldwide web, we must know first that we need the information, then its name. No Open Sesame! reveals what we need to know and no more. We trust some information: Experience tells us how programmers are likely to use floatingpoint. Modern erroranalysis tells us how to enhance our prospects for success. It s more than merely a way for experts to validate ( we hope ) the software we distribute through prestigious numerical libraries like LAPACK and fdlibm. Erroranalysis tells us how to design floatingpoint arithmetic, like IEEE Standard 754, moderately tolerant of wellmeaning ignorance among programmers though not yet among programming language designers and implementors. July 30, :28 pm Work in Progress Subject to Supersession Page 8
9 Java has evolved How Java s FloatingPoint Hurts Everyone Everywhere from a small language targeted towards TVsettop boxes and networked toasterovens to a large language and operating system targeted towards to challenge Microsoft s hegemony. Everybody Everything Everywhere Microsoft is vulnerable because its flaky Windows system is not one system but many. Wouldbe vendors of software for MS Windows have to cope with innumerable versions, a legacy of partially corrected bugs, unresolved incompatibilities,. Software often fails to install or later malfunctions because diversity among Windows systems has become unmanageable by the smaller software developers who cannot afford to pretest their work upon every kind of Windows system. Java s Write Once, Run Anywhere tantalizes software vendors with the prospect of substantially less debugging and testing than they have had to undertake in the past. This prospect has been invoked spuriously to rationalize Java s adherence to bad floatingpoint design decisions that mattered little in Java s initial niche market but now can t be reconciled with Java s expanded scope. Later we shall see why Java s expanded market would be served better by actual conformity to the letter and spirit of IEEE Standard 754 for Binary FloatingPoint Arithmetic. July 30, :28 pm Work in Progress Subject to Supersession Page 9
10 Pure Java s Two Cruel Delusions: Write Once, Run Anywhere and Linguistically Enforced Exact Reproducibility of all FloatingPoint Results These do figure among ideals that should influence our decisions. So does Universal Peace. But some ideals are better approached than reached, and best not approached too directly. ( How do you feel about Universal Death as a direct approach to Universal Peace? ) Pure Java s two cruel delusions are inconsistent with three facts of computing life: RushtoMarket engenders mistakes, bugs, versions, incompatibilities, conflicts, as in Java s oft revised AWT ( Window interface ), disputes between Sun and Microsoft,. Intentionally and unintentionally divergent implementations of the JVM will exist inevitably. Compliance with standards that reinforce commercial disparities can be enforced only by the kind of power to punish heretics for which emperors and popes used to yearn. JavaSoft lacks even the power to prevent heretic versions of Java from becoming preponderant in some markets. A healthy balance between Stability and Progress requires an approach to the Management of Change more thoughtful than can be expected from business entities battling for market share. Perfect uniformity and stability, if taken literally, are promises beyond Java s power to fulfill. Suppose for argument s sake that the two cruel delusions were not delusions. Suppose they became actuality at some moment in time. This situation couldn t last long. To understand why consider Complex Arithmetic Classes. July 30, :28 pm Work in Progress Subject to Supersession Page 10
11 Complex Arithmetic Classes. Why More than One? JavaSoft would promulgate its 100% Pure Java Complex Arithmetic Class Library, and the Free Software Foundation would promulgate another ( you d have to install it yourself ), and the Regents of the University of California would offer Kahan s Complex Arithmetic Class Library. How would Kahan s differ from JavaSoft s? In line with the C9X proposal before ANSI X3J11, he includes an Imaginary Class and allows complex variables to be written as x + ı*y or x + y*ı ( where ı := ( 1) is the declared imaginary unit ) instead of sticking to Fortranlike (x, y) as James Gosling has proposed. Kahan s imaginary class allows real and complex to mix without forcing coercions of real to complex. Thus his classes avoid a little wasteful arithmetic ( with zero imaginary parts ) that compilers can have trouble optimizing away. Other than that, with overloaded infix arithmetic operators, you can t tell the difference between Kahan s syntax and Gosling s. Imagine now that you are developing software intended to work upon your customer s Complex functions, perhaps to compute their contour integrals numerically and to plot them in interesting ways. Can you assume that your market will use only JavaSoft s Complex classes? Why should you have to test your software s compatibility with all the competing Complex classes? Wouldn t you rather write just once, debug just once, and then run anywhere that the official Pure JavaSoft Complex Classes are in use, and ignore potential customers who use those heretic alternatives? But some heresies cannot be ignored. July 30, :28 pm Work in Progress Subject to Supersession Page 11
12 Example: Borda s Mouthpiece, a classical two dimensional fluid flow Define complex analytic functions g( z) = z 2 + z z 2 + 1, and Fz () = 1+ gz () +log ( gz ()). Plot the values taken by F(z) as complex variable z runs along eleven rays z = r i, z = r e 4i π/10, z = r e 3i π/10, z = r e 2i π/10, z = r e i π/10, z = r and their Complex Conjugates, taking positive r from near 0 to near +. These rays are streamlines of an ideal fluid flowing in the right halfplane into a sink at the origin. The left halfplane is filled with air flowing into the sink. The vertical axis is a free boundary; its darker parts are walls inserted into the flow without changing it. The function F(z) maps this flow conformally to a flow with the sink moved to and the walls, pivoting around their innermost ends, turned into the left halfplane but kept straight to form the parallel walls of a long channel. ( Perhaps the Physics is idealized excessively, but that doesn t matter here.) The expected picture, Borda s Mouthpiece, should show eleven streamlines of an ideal fluid flowing into a channel under pressure so high that the fluid s surface tears free from the inside of the channel. July 30, :28 pm Work in Progress Subject to Supersession Page 12
13 Borda s Mouthpiece Correctly plotted Streamlines Streamlines should not cut across each other! 5 5 Y( I, U ) 0 y( I, U ) X( I, U ) x( I, U) Plotted using C9X like Complex and Imaginary Misplotted using Fortran like Complex An Ideal Fluid under high pressure escapes to the left through a channel with straight horizontal sides. Inside the channel, the flow's boundary is free, it does not touch the channel walls. But when 0 is mishandled, as Fortranstyle Complex arithmetic must mishandle it, that streamline of the flow along and underneath the lower channel wall is misplotted across the inner mouth of the channel and, though it does not show above, also as a short segment in the upper wall at its inside end. Both plots come from the same program using different Complex Class libraries, first with and second without an Imaginary Class. July 30, :28 pm Work in Progress Subject to Supersession Page 13
14 Lifting Flow past Joukowski s Aerofoil Correctly Plotted Streamlines Where is this wing s bottom? Plotted using C9X like Complex and Imaginary Misplotted using Fortran like Complex A circulating component, necessary to generate lift, speeds the flow of an idealized fluid above the wing and slows it below. One streamline splits at the wing s leading edge and recombines at the trailing edge. But when 0 is mishandled, as Fortranstyle Complex arithmetic must mishandle it, that streamline goes only over the wing. The computation solves numerically nontrivial transcendental equations involving complex logarithms. Both plots come from the same program using different Complex Class libraries, first with and second without an Imaginary Class. Experienced practitioners programming in Fortran or C++ have learned to replace the split streamline by two streamlines, one above and one below, separated by as few rounding errors as produce a goodlooking plot. July 30, :28 pm Work in Progress Subject to Supersession Page 14
15 Why such plots malfunction, and a very simple way to correct them, were explained long ago in Branch Cuts for Complex Elementary Functions, or Much Ado About Nothing's Sign Bit by W. Kahan, ch. 7 in The State of the Art in Numerical Analysis ( 1987 ) ed. by M. Powell and A. Iserles for Oxford U.P. A streamline goes astray when the complex functions SQRT and LOG are implemented, as is necessary in Fortran and in libraries currently distributed with C/C++ compilers, in a way that disregards the sign of ± 0.0 in IEEE 754 arithmetic and consequently violates identities like SQRT( CONJ( Z ) ) = CONJ( SQRT( Z ) ) and LOG( CONJ( Z ) ) = CONJ( LOG( Z ) ) whenever the COMPLEX variable Z takes negative real values. Such anomalies are unavoidable if Complex Arithmetic operates on pairs (x, y) instead of notional sums x + ı y of real and imaginary variables. The language of pairs is incorrect for Complex Arithmetic; it needs the Imaginary type. A controversial Complex Arithmetic Extension to the programming language C incorporating that correction, among other things, has been put before ANSI X3J11, custodian of the C language standard, as part of the C9X proposal. It is controversial because it purports to help programmers cope with certain physically important discontinuities by suspending thereat ( and nowhere else ) the logical proposition that x == y implies f(x) == f(y). Many a programmer will prefer this anomaly to its alternatives The moral of this story: There will always be good reasons ( and bad ) to call diverse versions of hard and software, including mathematical software, by the same name. Nobody can copyright Complex Class. July 30, :28 pm Work in Progress Subject to Supersession Page 15
16 Besides programs with the same name but designed for slightly different results, there are programs with the same name designed to produce essentially the same results as quickly as possible which must therefore produce slightly different results on different computers. Roundoff causes results to differ slightly not because different computers round arithmetic differently but because they manage memory, caches and register files differently. Example: Matrix multiplication C := A B i.e. c ij := k a ik b kj = a i1 b 1j + a i2 b 2j + a i3 b 3j + To keep pipelines full and avoid unnecessary cache misses, different computer architectures have to perform multiplications a ik b kj and their subsequent additions in different orders. In the absence of roundoff the order would not affect C because addition would be associative. Order affects accuracy only a little in the presence of roundoff because, for all suitable matrix norms, C  A B /( A B ) cannot much exceed the roundoff threshold regardless of order, and this constraint upon C suffices for most applications even if C varies very noticeably from one computer to another. Ordering affects speed a lot. On most processors today, the most obvious matrix multiply program runs at least three times slower than a program with optimal blocking and loopunrolling. Optimization depends delicately upon processor and cache details. For matrices of large dimensions, a code optimized for an UltraSPARC, about three times faster thereon than an unoptimized code, runs on a Pentium Pro ( after recompilation ) slower than a naive code and about six times slower than its optimal code. Speed degradation becomes worse on multiprocessors. Faster matrix multiplication is usually too valuable to forego for unneeded exact reproducibility. Conclusion: Linguistically legislated exact reproducibility is unenforceable. July 30, :28 pm Work in Progress Subject to Supersession Page 16
17 The merely Difficult we do immediately; the Impossible will take slightly longer. Royal Navy maxim adopted during WW II by American Seabees. Everincreasing diversity in hardware and software compounds the difficulty of testing new software intended for the widest possible market. Soon Difficult must become Impossible unless the computing industry collectively and programmers individually share a burden of SelfDiscipline: Modularize designs, so that diversity will add to your testing instead of multiplying it. Know your market, or target only the markets you know; exploit only capabilities you know to be available in all of your targeted markets. Eliminate needless diversity wherever possible, though this is easier said than done; Things should be as simple as possible, but no simpler. Albert Einstein. Java s designers, by pursuing the elimination of diversity beyond the point of oversimplification, have turned a very desirable design goal into an expendable fetish. They have mixed up two ideas: Exact Reproducibility, needed by some floatingpoint programmers sometimes, and Predictability within Controllable Limits, needed by all programmers all the time. By pushing Exact Reproducibility of FloatingPoint to an illogical extreme, the designers ensure it will be disparaged, disregarded and finally jettisoned, perhaps carrying Predictability away too in the course of a Business Decision that could all too easily achieve what the British call Throwing Baby out with the bath water. July 30, :28 pm Work in Progress Subject to Supersession Page 17
18 The essence of programming is Control. Control requires Predictability, which should be Java s forte. Java would impose Exact Reproducibility upon FloatingPoint to make it Predictable. But Exact Reproducibility is JavaSoft s euphemism for Do as Sun s SPARCs do. Thus it denies programmers the choice of better floatingpoint running on most other hardware. Denied better choices, the programmer is not exercising Control but being controlled. Throwing Baby out with the bath water: When Exact Reproducibility of floatingpoint becomes too burdensome to implementors whose first priority is high speed, they will jettison Exact Reproducibility and, for lack of sound guidance, they will most likely abandon Predictability along with it. That s happening now. That s what Gosling s Loose Numerics amounts to; a better name for it is Sloppy Numerics. To achieve FloatingPoint Predictability: Limit programmers choices to what is reasonable and necessary as well as parsimonious, and Limit language implementors choices so as always to honor the programmer s choices. To do so, language designers must understand floatingpoint well enough to validate their determination of what is reasonable and necessary, or else must entrust that determination to someone else with the necessary competency. But Java s designers neglected timely engagement of Sun s inhouse numerical expertise, which would have prevented their floatingpoint blunders. Footnote: Validate a programming language s design? The thought appalls people who think such design is a Black Art. Many people still think FloatingPoint is a Black Art. They are wrong too. July 30, :28 pm Work in Progress Subject to Supersession Page 18
19 Java purports to fix what ain t broken in Floatingpoint. Floatingpoint arithmetic hardware conforming to IEEE Standard 754, as does practically all today s commercially significant hardware on desktops, is already among the least diverse things, hard or software, so ubiquitous in computers. Now Java, mistakenly advertised as conforming to IEEE 754 too, pretends to lessen its diversity by adding another one to the few extant varieties of floatingpoint. How many significantly different floatingpoint hardware architectures matter today? Four : #0: Signal processors that may provide float and/or floatextended but not double. #1: RISCbased computers that provide 4byte float and 8byte double but nothing wider. #2: PowerPC; MIPS R10000; HP 8000 : same as #1 plus fused multiplyadd operation. #3: Intel x86, Pentium; clones by AMD and Cyrix; Intel 80960KB; new Intel/HP IA64; and Motorola 680x0 and : the same as #1 plus a 10+byte long double. Over 95% of the computers on desktops have architecture #3. Most of the rest have #2. Both #3 and #2 can be and are used in restricted ways that match #1 as nearly as matters. All of #1, #2, #3 support Exception Flags and Directed Roundings, capabilities mandated by IEEE Standard 754 but generally omitted from architecture #0 because they have little value in its specialized market. Java would add a fifth floatingpoint architecture #0.5 between #0 and #1. It omits from architecture #1 the Exception Flags and Directed Roundings IEEE 754 requires. July 30, :28 pm Work in Progress Subject to Supersession Page 19
20 Java linguistically confuses the issues about floatingpoint Exceptions: Java, like C++, misuses the word Exception to mean what IEEE 754 calls a Trap. Java has no words for the five floatingpoint Events that IEEE 754 calls Exceptions : Invalid Operation, Overflow, DivisionbyZero, Underflow, Inexact Result These events are not errors unless they are handled badly. They are called Exceptions because to any policy for handling them, imposed in advance upon all programmers by the computer system, some programmers will have good reasons to take exception. IEEE 754 specifies a default policy for each exception, and allows system implementors the option of offering programmers an alternative policy, which is to Trap ( jump ) with specified information about the exception to a programmerselected traphandler. We shall not go into traps here; they would complicate every language issue without adding much more than speed, and little of that, to what flags add to floatingpoint programming. ( Borneo would provide some support for traps.) IEEE 754 specifies five flags, one named for each exception: Invalid Operation, Overflow, DivisionbyZero, Underflow, Inexact Result A flag is a type of global variable raised as a sideeffect of exceptional floatingpoint operations. Also it can be sensed, saved, restored and lowered by a program. When raised it may, in some systems, serve an extralinguistic diagnostic function by pointing to the first or last operation that raised it. Java lacks these flags and cannot conform to IEEE 754 without them. July 30, :28 pm Work in Progress Subject to Supersession Page 20
21 Invalid Operation, Overflow, DivisionbyZero, Underflow, Inexact Result IEEE 754 specifies a default policy for each of these kinds of floatingpoint exception: ı Signal the event by raising an appropriate one of the five flags, if it has not already been raised. ıı (Pre)substitute a default value for what would have been the result of the exceptional operation: Name of Flag and Exception Invalid Operation Overflow DivisionbyZero (Pre)substituted Default Value NotaNumber (NaN), which arithmetic propagates; or a huge integer on overflowed flt.pt. integer conversion ± approximately, depending on Rounding Direction ± Infinity exactly from finite operands. Underflow Inexact Result Gradual Underflow to a Subnormal (very tiny) value Rounded or Over/Underflowed result as usual ııı Resume execution of the program as if nothing exceptional had occurred. With these default values, IEEE 754 s floatingpoint becomes an Algebraically Completed system; this means the computer s every algebraic operation produces a welldefined result for all operands. Why should computer arithmetic be Algebraically Completed? What s wrong with the Defaults specified for these Exceptions by IEEE 754? Why does IEEE 754 specify a flag for each of these kinds of exception? The next three pages answer these three questions and a fourth: What should Java do?. July 30, :28 pm Work in Progress Subject to Supersession Page 21
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